Diffusion in an incompressible random flow (Q1128364)

The equation \[ {{\partial \rho}\over{\partial t}} =\Delta \rho - b(x) \nabla \rho , \quad x \in \mathbb{R}^d , t \geq 0 \] where \( b(x) \) is a random homogeneous solenoidal vector field in \(\mathbb{R}^d \) with zero mean, which describes the distribution density of a Brownian particle in a stationary fluid flow, is considered. The asymptotic behaviour of the density \( \rho (x,t) \) for large values of \( t \) is investigated. It is pointed out that the behaviour of the particle is diffusive when the effect of the convective term \( b\nabla \rho \) is reduced to the replacement of the original equation to some ``homogenized'' equation. Relationship between homogenization and diffusive behaviour is established.